Rhombic triacontahedron

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Rhombic triacontahedron
Rhombic triacontahedron
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TypeCatalan
Face polygonrhombus
Faces30
Edges60
Vertices32 = 20 + 12
Face configuration3,5,3,5
Symmetry groupicosahedral (Ih)
Dual polyhedronicosidodecahedron
Propertiesconvex, face/edge-uniform, zonohedron

In geometry, the rhombic triacontahedron is a convex polyhedron with 30 rhombic faces. It is the polyhedral dual of the icosidodecahedron, and it is a zonohedron. The ratio of the long diagonal to the short diagonal of each face is exactly equal to the golden ratio, φ, so that the acute angles on each face measure 2 tan−1(1/φ), or approximately 63.43°.

Being the dual of an Archimedean polyhedron, the rhombic triacontahedron is face-uniform, meaning the symmetry group of the solid acts transitively on the set of faces. In elementary terms, this means that for any two faces A and B there is a rotation or reflection of the solid that leaves it occupying the same region of space while moving face A to face B. The rhombic triacontahedron is also somewhat special in being one of the nine edge-uniform convex polyhedra, the others being the five Platonic solids, the cuboctahedron, the icosidodecahedron, and the rhombic dodecahedron.

The rhombic triacontahedron forms the (hull of) the projection of a 6-dimensional hypercube to 3 dimensions.

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